The KR rearrangement between a pair of positive
densities is a triangular bijection on
\(\mathbb {R}^n\) 
such that each slice (REF ) is
strictly increasing and absolutely continuous
[1]}.
Hence, it makes sense to consider candidate
rearrangements \(U\)  whose \(k\) th components, \(U^k\) , can
be written as
\( U^k(z_1,\ldots ,z_k)= a_k(z_1,\ldots ,z_{k-1}) +\int _0^{z_k} \exp \left( b_k(z_1,\ldots ,z_{k-1}, t) \right)\, {\rm d} t,\) 
